How to simplify the proof of right-properness?

Question

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback is always a weak equivalence). Or is there a close statement that I can't remember (browsing nLab did not help me) ?

Comments. Consider the diagram $\mathbf{D}=X\rightarrow Y \leftarrow Z$ where the left-hand map is a fibration and the right-hand map a weak equivalence. Let $T=\projlim \mathbf{D}$. Choose a fibrant replacement $Y^{fib}$ for $Y$ and a trivial cofibration $Y\rightarrow Y^{fib}$. Factor the composite map $X \rightarrow Y \rightarrow Y^{fib}$ as a composite trivial cofibration-fibration. We obtain a diagram $\mathbf{E}=X^{fib}\rightarrow Y^{fib}\leftarrow Z$ such that the left-hand map is a fibration between fibrant objects and the right-hand map is a weak equivalence. Let $U=\projlim \mathbf{E}$. By hypothesis, the map $U\rightarrow X^{fib}$ is a weak equivalence. By construction, the map of diagrams $\mathbf{D} \rightarrow \mathbf{E}$ is a weak equivalence of diagrams. The map $T\rightarrow X$ is a weak equivalence iff the map $T\rightarrow U$ is a weak equivalence. What next ?

Why. I found this cryptic remark in my notebook, and I can't remember where it comes from. The reason why I want to simplify the proof of right properness is that I have to deal with model categories where a set of generating trivial cofibrations is not known. I only know what I call a set of generating anodyne cofibrations. And the trivial fibrations which are the anodyne fibrations (i.e. having the RLP with respect to the set of generating anodyne cofibrations) which are a dual strong deformation retract. And the reason why I am interested in right properness is that I want to study right Bousfield localizations.

Answer 1

To complete the argument you need to apply K. Brown's Lemma. Call your model category $\mathcal{M}$, then the map $Z \to Y$ induces a pullback functor $\mathcal{M} \downarrow Y \to \mathcal{M} \downarrow Z$ and the lemma implies that it preserves weak equivalences between fibrations over $Y$. If you define $V \to Y$ as the pullback of $X^{\mathrm{fib}} \to Y^{\mathrm{fib}}$, then $X \to V$ is a weak equivalence by the assumption and 2-out-of-3. Then apply the previous observation to $X \to V$ to see that $T \to U$ is a also weak equivalence and hence so is $T \to X$.

A reference for this is Lemma 9.4 in Bousfield's On the Telescopic Homotopy Theory of Spaces, but I imagine it was known well before that.

Answer 2

You asked if you can check it for less than the collection of all weak equivalences. In A.5 of Motivic Symmetric Spectra, Jardine proves a general right properness result from Corollary A.4, which is the statement that weak equivalences with fibrant codomain are preserved by pullback along a fibration. So that appears to answer your question. His proof of Corollary A.4 uses facts specific to his setting, but it's still quite general.

Since you mentioned maps dual to strong deformation retracts, I feel like I should also advertize Rezk's work. Rezk calls a map $f : X \to Y$ "sharp" if for each base-change of f along any map into the base Y the resulting pullback square is homotopy cartesian. These sharp maps are dual to the flat maps Hopkins invented, which appear in the appendix of the Kervaire paper and which appear in other works (e.g. Batanin-Berger) being called h-cofibrations. I think of these flat maps like strong deformation retracts, and that's why I thought you might want to hear about sharp maps.

In Proposition 2.2 of "Fibrations and homotopy colimits of simplicial sheaves", Rezk proves that a model category is right proper if and only if every fibration is sharp. This same result appeared in Morel Voevodsky when they were trying to prove the motivic category was right proper (it was a long proof, but this fact was crucial). Perhaps this will also help you carry out your proof.